Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
413 conjugacy classes of subgroups; only the 92 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 92 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1895 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C2 | #9078 |
| prime radical | #3067 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #17336 | C2 | semidirect |
| 3. #12792 | C2 | semidirect |
| 4. #11604 | C2 | semidirect |
| 5. C2×#1336 | C2 | semidirect |
| 6. #7182 | C2 | semidirect |
| 7. #7181 | C2 | semidirect |
| 101. C2×Q16 | M16 | semidirect |
| 103. C2×D8 | M16 | semidirect |
| 148. #108 | M16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21558 | 2 |
| #17336 | 2 |
| #12792 | 2 |
| #11604 | 2 |
| C2×#1336 | 2 |
| #7182 | 2 |
| #7181 | 2 |